## Eigenanalysis

### At which station am I?

An adjacency matrix represents graph topology (how the nodes are connected). However, a matrix can not only represent the connections, but also can be applied to a vector and can generate a new vector. We saw the adjacency matrix can generate a new station vector. We can continue this computation a bit more. Why do we this? I want to show you a bit interesting stuffs. Let's assume we are first at Weinmeisterstr. We repeat visiting to next station or staying the station. Each computation result shows how many possible paths are there to reach each station. Let's see how this number goes.

This is initial state, we are at Weinmeisterstr station.
\begin{eqnarray*} \begin{array}{|c|c|} \hline \mbox{Station} & \mbox{1 step} \\ \hline \mbox{Weinmeisterstr} & 1 \\ \mbox{Alexanderplatz} & 0 \\ \mbox{Hackescher Markt} & 0 \\ \mbox{Jannowitzbruecke} & 0 \\ \hline \end{array} \end{eqnarray*} Second step result is:

\begin{eqnarray*} \begin{array}{|c|c|} \hline \mbox{Station} & \mbox{2 steps} \\ \hline \mbox{Weinmeisterstr} & 2 \\ \mbox{Alexanderplatz} & 2 \\ \mbox{Hackescher Markt} & 1 \\ \mbox{Jannowitzbruecke} & 1 \\ \hline \end{array} \end{eqnarray*}

Let's continue to 3 steps, 5 steps, and 10 steps.
\begin{eqnarray*} \begin{array}{|c|c|c|c|} \hline \mbox{Station} & \mbox{3 steps} & \mbox{4 steps} & \mbox{10 steps} \\ \hline \mbox{Wein.} & 4 & 26 & 3862 \\ \mbox{Alex.} & 6 & 44 & 6688 \\ \mbox{Hack.} & 3 & 25 & 3861 \\ \mbox{Jann.} & 3 & 25 & 3861 \\ \hline \end{array} \end{eqnarray*}
I guess the number of paths to be Alexanderplatz seems twice larger than to be other stations. I think this kind of guess is important in mathematics. I enjoy mathematics by finding a pattern.

Actually, it is not a coincidence that the possible number of paths to go to Alexanderplatz is twice to the other stations when number steps is large. It is interesting to me that such a pattern is here, this would have been more chaotic. The following sections, I would like to look into more details. However, I hope you have some kind of feeling that we can analyze the relationship using graph theory and adjacency matrix.

I would like to come back to the first question. That is How can we analyze the relationship of authors''. The mathematics we see here consider only relationship between objects. We see the examples of numbers, authors, and train stations. But graph theory and adjacency matrix don't care what are the objects. Figures 4, 5, 6 are all the same graph and shares the same adjacency matrix. You might think this is careless or inhuman. But on the other hand, don't care means anything can fit this. We can use these tools (graph theory and adjacency matrix) not only for train stations, but also for authors. This makes these tools powerful and versatile. We see we can compute how to visit the train stations like from station A to station B. We can also compute how to reach the person like from person A to person B. We can find a pattern in relationships.
 Figure 4:Edges connect nodes.
 Figure 5. The same graphs. Graph only cares the connections between nodes.
 Figure 6: Graph example 1. Each node is an English author.
In the next article, we will see a method of finding a feature of adjacency matrix''.